Wireless Networking and Communications Group IMPROVING WIRELESS DATA

Wireless Networking and Communications Group IMPROVING WIRELESS DATA TRANSMISSION SPEED AND RELIABILITY TO MOBILE COMPUTING PLATFORMS Prof. Brian L. Evans Lead Graduate Students Aditya Chopra, Kapil Gulati and Marcel Nassar In collaboration with Keith R. Tinsley and Chaitanya Sreerama at Intel Labs 12/15/2021 Texas Wireless Summit, Austin, Texas

Problem Definition 2 Within computing platforms, wireless transceivers experience radio frequency interference (RFI) from clocks and busses Objectives Develop offline methods to improve communication performance in presence of computer platform RFI Develop adaptive online algorithms for these methods Approach We will use noise and interference interchangeably Statistical modeling of RFI Filtering/detection based on estimated model parameters Wireless Networking and Communications Group

Common Spectral Occupancy 3 Standard Band (GHz) Wireless Networking Interfering Clocks and Busses Bluetooth 2. 4 Personal Area Network Gigabit Ethernet, PCI Express Bus, LCD clock harmonics IEEE 802. 11 b/g/n 2. 4 IEEE 802. 16 e 2. 5– 2. 69 3. 3– 3. 8 5. 725– 5. 85 Mobile Broadband (Wi-Max) PCI Express Bus, LCD clock harmonics IEEE 802. 11 a 5. 2 Wireless LAN (Wi -Fi) PCI Express Bus, LCD clock harmonics Wireless LAN (Wi Gigabit Ethernet, PCI Express Bus, -Fi) LCD clock harmonics Wireless Networking and Communications Group

Impact of RFI 4 Impact of LCD noise on throughput performance for a 802. 11 g embedded wireless receiver [Shi et al. , 2006] Wireless Networking and Communications Group Backup

Our Contributions 5 Mitigation of computational platform noise in single carrier, single antenna systems [Nassar et al. , ICASSP 2008] Computer Platform Noise Modelling Evaluate fit of measured RFI data to noise models Narrowband Interference: Middleton Class A model Broadband Interference: Symmetric Alpha Stable Parameter Estimation Evaluate estimation accuracy vs complexity tradeoffs Filtering / Detection Evaluate communication performance vs complexity tradeoffs • Middleton Class A: Correlation receiver, Wiener filtering and Bayesian detector • Symmetric Alpha Stable: Myriad filtering, hole punching, and Bayesian detector Wireless Networking and Communications Group

Power Spectral Densities Middleton Class A Parameter values: A = 0. 15 and = 0. 1 Symmetric Alpha Stable Parameter values: = 1. 5, = 0 and = 10

Fitting Measured RFI Data 7 Broadband RFI data 80, 000 samples collected using 20 GSPS scope Backup Estimated Parameters Symmetric Alpha Stable Model Localization (δ) 0. 0043 Characteristic exp. (α) 1. 2105 Dispersion (γ) 0. 2413 Distance 0. 0514 Middleton Class A Model Overlap Index (A) 0. 1036 Gaussian Factor (Γ) 0. 7763 Distance 0. 0825 Gaussian Model Wireless Networking and Communications Group Mean (µ) 0 Variance (σ2) 1 Distance: Kullback-Leibler divergence Distance 0. 2217

Fitting Measured RFI Data Best fit for other 25 data sets under different conditions Return

Filtering and Detection Methods 9 Middleton Class A noise Symmetric Alpha Stable noise Filtering Wiener Filtering (Linear) Backup [Gonzalez & Arce, 2001] Detection Correlation Receiver (Linear) MAP approximation Backup Small Signal Approximation to MAP detector [Spaulding & Middleton, 1977] Hole Punching Backup Detection MAP (Maximum a posteriori probability) detector [Spaulding & Middleton, 1977] Myriad Filtering Backup Wireless Networking and Communications Group Backup

Results: Class A Detection 10 Pulse shape Raised cosine 10 samples per symbol 10 symbols per pulse Method Wireless Networking and Communications Group Complexity Channel A = 0. 35 = 0. 5 × 10 -3 Memoryless Detection Perform. Correl. Low Wiener Medium Low MAP Approx. Medium High MAP High

Results: Alpha Stable Detection 11 Backup Method Complexity Detection Perform. Hole Punching Low Medium Selection Myriad Low Medium MAP Approx. Medium High Optimal Myriad High Medium Backup Use dispersion parameter in place of noise variance to generalize SNR Wireless Networking and Communications Group

Results: Class A for 2 2 MIMO 12 Improvement in communication performance over conventional Gaussian ML receiver at symbol error rate of 10 -2 Complexity Analysis A Noise Characteristic Improve -ment 0. 01 Highly Impulsive ~15 d. B 0. 1 Moderately Impulsive ~8 d. B Nearly Gaussian ~0. 5 d. B 1 Communication Performance (A = 0. 1, 1= 0. 01, 2= 0. 1, k = 0. 4) Wireless Networking and Communications Group

Conclusions 13 Radio frequency interference from computing platform Affects wireless data communication transceivers Fit Middleton Class A and symmetric alpha stable models RFI mitigation can reduce bit error rate by a factor of 100 for Middleton Class A model, single carrier system 10 for Middleton Class A model, 2 x 2 MIMO system 10 for Symmetric Alpha Stable model, single carrier system Other applications of impulsive noise models Co-channel interference Adjacent channel interference Wireless Networking and Communications Group

Contributions 14 Publications M. Nassar, K. Gulati, A. K. Sujeeth, N. Aghasadeghi, B. L. Evans and K. R. Tinsley, “Mitigating Nearfield Interference in Laptop Embedded Wireless Transceivers”, Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc. , Mar. 30 -Apr. 4, 2008, Las Vegas, NV USA. K. Gulati, A. Chopra, R. W. Heath Jr. , B. L. Evans, K. R. Tinsley, and X. E. Lin, ”MIMO Receiver Design in the Presence of Radio Frequency Interference”, Proc. IEEE Int. Global Communications Conf. , Nov. 30 -Dec. 4 th, 2008, New Orleans, LA USA, accepted for publication. A. Chopra, K. Gulati, B. L. Evans, K. R. Tinsley, and C. Sreerama, ``Performance Bounds of MIMO Receivers in the Presence of Radio Frequency Interference'', Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc. , Apr. 19 -24, 2009, Taipei, Taiwan, submitted. Software Releases RFI Mitigation Toolbox Version 1. 1 Beta (Released November 21 st, 2007) Version 1. 0 (Released September 22 nd, 2007) Project Website http: //users. ece. utexas. edu/~bevans/projects/rfi/index. html Wireless Networking and Communications Group

15 Thank You, Questions ? Wireless Networking and Communications Group

References 16 RFI Modeling [1] D. Middleton, “Non-Gaussian noise models in signal processing for telecommunications: New methods and results for Class A and Class B noise models”, IEEE Trans. Info. Theory, vol. 45, no. 4, pp. 1129 -1149, May 1999. [2] K. F. Mc. Donald and R. S. Blum. “A physically-based impulsive noise model for array observations”, Proc. IEEE Asilomar Conference on Signals, Systems& Computers, vol 1, 2 -5 Nov. 1997. [3] K. Furutsu and T. Ishida, “On theory of amplitude distributions of impulsive random noise, ” J. Appl. Phys. , vol. 32, no. 7, pp. 1206– 1221, 1961. [4] J. Ilow and D. Hatzinakos, “Analytic alpha-stable noise modeling in a Poisson field of interferers or scatterers”, IEEE transactions on signal processing, vol. 46, no. 6, pp. 1601 -1611, 1998. Parameter Estimation [5] S. M. Zabin and H. V. Poor, “Efficient estimation of Class A noise parameters via the EM [Expectation-Maximization] algorithms”, IEEE Trans. Info. Theory, vol. 37, no. 1, pp. 60 -72, Jan. 1991 [6] G. A. Tsihrintzis and C. L. Nikias, "Fast estimation of the parameters of alpha-stable impulsive interference", IEEE Trans. Signal Proc. , vol. 44, Issue 6, pp. 1492 -1503, Jun. 1996 RFI Measurements and Impact [7] J. Shi, A. Bettner, G. Chinn, K. Slattery and X. Dong, "A study of platform EMI from LCD panels impact on wireless, root causes and mitigation methods, “ IEEE International Symposium on Electromagnetic Compatibility, vol. 3, no. , pp. 626 -631, 14 -18 Aug. 2006 Wireless Networking and Communications Group

References (cont…) 17 Filtering and Detection [8] A. Spaulding and D. Middleton, “Optimum Reception in an Impulsive Interference Environment. Part I: Coherent Detection”, IEEE Trans. Comm. , vol. 25, no. 9, Sep. 1977 [9] A. Spaulding and D. Middleton, “Optimum Reception in an Impulsive Interference Environment Part II: Incoherent Detection”, IEEE Trans. Comm. , vol. 25, no. 9, Sep. 1977 [10] J. G. Gonzalez and G. R. Arce, “Optimality of the Myriad Filter in Practical Impulsive-Noise Environments”, IEEE Trans. on Signal Processing, vol 49, no. 2, Feb 2001 [11] S. Ambike, J. Ilow, and D. Hatzinakos, “Detection for binary transmission in a mixture of Gaussian noise and impulsive noise modelled as an alpha-stable process, ” IEEE Signal Processing Letters, vol. 1, pp. 55– 57, Mar. 1994. [12] J. G. Gonzalez and G. R. Arce, “Optimality of the myriad filter in practical impulsive-noise environments, ” IEEE Trans. on Signal Proc, vol. 49, no. 2, pp. 438– 441, Feb 2001. [13] E. Kuruoglu, “Signal Processing In Alpha Stable Environments: A Least Lp Approach, ” Ph. D. dissertation, University of Cambridge, 1998. [14] J. Haring and A. J. Han Vick, “Iterative Decoding of Codes Over Complex Numbers for Impulsive Noise Channels”, IEEE Trans. On Info. Theory, vol 49, no. 5, May 2003 [15] Ping Gao and C. Tepedelenlioglu. “Space-time coding over mimo channels with impulsive noise”, IEEE Trans. on Wireless Comm. , 6(1): 220– 229, January 2007. Wireless Networking and Communications Group

Backup Slides 18 Most backup slides are linked to the main slides Miscellaneous topics not covered in main slides Performance bounds for single carrier single antenna system in presence of RFI Backup Wireless Networking and Communications Group

Outline 1 9 Problem definition Single carrier single antenna systems Radio frequency interference modeling Estimation of interference model parameters Filtering/detection Multi-input multi-output (MIMO) single carrier systems Conclusions Wireless Networking and Communications Group

Impact of RFI 20 Calculated in terms of desensitization (“desense”) Return Interference raises noise floor Receiver sensitivity will degrade to maintain SNR Desensitization levels can exceed 10 d. B for 802. 11 a/b/g due to computational platform noise [J. Shi et al. , 2006] Case Sudy: 802. 11 b, Channel 2, desense of 11 d. B More than 50% loss in range Throughput loss up to ~3. 5 Mbps for very low receive signal strengths (~ -80 dbm) Wireless Networking and Communications Group

Statistical Modeling of RFI 2 1 Radio Frequency Interference (RFI) Sum of independent radiation events Predominantly non-Gaussian impulsive statistics Key Statistical-Physical Models Middleton Class A, B, C models Independent of physical conditions (Canonical) Sum of independent Gaussian and Poisson interference Model non-linear phenomenon governing RFI Symmetric Alpha Stable models Approximation of Middleton Class B model Wireless Networking and Communications Group Backup

Assumptions for RFI Modeling 2 2 Key Assumptions [Middleton, 1977][Furutsu & Ishida, 1961] Infinitely many potential interfering sources with same effective radiation power Power law propagation loss Poisson field of interferers Pr(number of interferers = M |area R) ~ Poisson distributed emission times Temporally independent (at each sample time) Limitations [Alpha Stable]: Does not include thermal noise Temporal dependence may exist Wireless Networking and Communications Group

Middleton Class A, B and C Models 23 Return [Middleton, 1999] Class A Class B Class C Narrowband interference (“coherent” reception) Uniquely represented by 2 parameters Broadband interference (“incoherent” reception) Uniquely represented by six parameters Sum of Class A and Class B (approx. Class B) Wireless Networking and Communications Group Backup

Middleton Class A model 2 4 Probability Density Function PDF for A = 0. 15, = 0. 8 Parameter Description Range Overlap Index. Product of average number of emissions per A [10 -2, 1] second and mean duration of typical emission Gaussian Factor. Ratio of second-order moment of Gaussian Γ [10 -6, 1] component to that of non-Gaussian component Wireless Networking and Communications Group

Middleton Class B Model 25 Envelope Statistics Envelope exceedence probability density (APD), which is 1 – cumulative distribution function (CDF) Wireless Networking and Communications Group Return

Middleton Class B Model (cont…) 26 Middleton Class B Envelope Statistics Wireless Networking and Communications Group Return

Middleton Class B Model (cont…) 27 Parameters for Middleton Class B Model Parameters Description Return Typical Range Impulsive Index AB [10 -2, 1] Ratio of Gaussian to non-Gaussian intensity ΓB [10 -6, 1] Scaling Factor NI [10 -1, 102] Spatial density parameter α [0, 4] Effective impulsive index dependent on α A α [10 -2, 1] Inflection point (empirically determined) εB > 0 Wireless Networking and Communications Group

Accuracy of Middleton Noise Models 28 ε 0 (d. B > εrms) Magnetic Field Strength, H (d. B relative to microamp per meter rms) Return P(ε > ε 0) Soviet high power over-the-horizon radar interference [Middleton, 1999] Wireless Networking and Communications Group Percentage of Time Ordinate is Exceeded Fluorescent lights in mine shop office interference [Middleton, 1999]

Symmetric Alpha Stable Model 29 Characteristic Function Closed-form PDF expression only for α = 1 (Cauchy), α = 2 (Gaussian), α = 1/2 (Levy), α = 0 (not very useful) Approximate PDF using inverse transform of power series expansion Second-order moments do not exist for α < 2 Generally, moments of order > α do not exist Backup Parameter PDF for = 1. 5, = 0 and = 10 Description Characteristic Exponent. Amount of impulsiveness Localization. Analogous to mean Dispersion. Analogous to variance Wireless Networking and Communications Group Backup Range

Symmetric Alpha Stable PDF 30 Closed form expression does not exist in general Power series expansions can be derived in some cases Standard symmetric alpha stable model for localization parameter = 0 Wireless Networking and Communications Group Return

Symmetric Alpha Stable Model 31 Heavy tailed distribution Return Density functions for symmetric alpha stable distributions for different values of characteristic exponent alpha: a) overall density and b) the tails of densities Wireless Networking and Communications Group

Estimation of Noise Model Parameters 32 Middleton Class A model Expectation Maximization (EM) [Zabin & Poor, 1991] Backup Find roots of second and fourth order polynomials at each iteration Advantage: Small sample size is required (~1000 samples) Disadvantage: Iterative algorithm, computationally intensive Symmetric Alpha Stable Model Based on Extreme Order Statistics [Tsihrintzis & Nikias, 1996] Backup Parameter estimators require computations similar to mean and standard deviation computations Advantage: Fast / computationally efficient (non-iterative) Disadvantage: Requires large set of data samples (~10000 samples) Wireless Networking and Communications Group

Parameter Estimation: Middleton Class A 33 Expectation Maximization (EM) Return E Step: Calculate log-likelihood function w current parameter values M Step: Find parameter set that maximizes log-likelihood function EM Estimator for Class A parameters [Zabin & Poor, 1991] Express envelope statistics as sum of weighted PDFs Maximization step is iterative Given A, maximize K (= A ). Root 2 nd order polynomial. Given K, maximize A. Root 4 th order polynomial Wireless Networking and Communications Group Results Backup

Expectation Maximization Overview 34 Return Wireless Networking and Communications Group

Results: EM Estimator for Class A 35 Return Normalized Mean-Squared Error in A Iterations for Parameter A to Converge K=A PDFs with 11 summation terms 50 simulation runs per setting Wireless Networking and Communications Group 1000 data samples Convergence criterion:

Results: EM Estimator for Class A 36 Return • For convergence for A [10 -2, 1], worstcase number of iterations for A = 1 • Estimation accuracy vs. number of iterations tradeoff Wireless Networking and Communications Group

Parameter Estimation: Symmetric Alpha Stable 37 Based on extreme order statistics [Tsihrintzis & Nikias, 1996] PDFs of max and min of sequence of i. i. d. data samples Return PDF of maximum PDF of minimum Extreme order statistics of Symmetric Alpha Stable PDF approach Frechet’s distribution as N goes to infinity Parameter Estimators then based on simple order statistics Advantage: Disadvantage: Fast/computationally efficient (non-iterative) Requires large set of data samples (N~10, 000) Results Wireless Networking and Communications Group Backup

Results: Symmetric Alpha Stable Parameter Estimator 38 Return • Data length (N) of 10, 000 samples • Results averaged over 100 simulation runs • Estimate α and “mean” directly from data • Estimate “variance” from α and δ estimates Mean squared error in estimate of characteristic exponent α Wireless Networking and Communications Group

Results: Symmetric Alpha Stable Parameter Estimator (Cont…) 39 Return Mean squared error in estimate of localization (“mean”) Wireless Networking and Communications Group Mean squared error in estimate of dispersion (“variance”)

Extreme Order Statistics 40 Return Wireless Networking and Communications Group

Parameter Estimators for Alpha Stable 41 Return 0<p<α Wireless Networking and Communications Group

Filtering and Detection 42 System Model Impulsive Noise Pulse Shapin g Matched Filter Detectio n Rule Assumptions: Pre. Filtering Multiple samples of the received signal are available N Path Diversity [Miller, 1972] N samples per symbol Oversampling by N [Middleton, 1977] Multiple samples increase gains vs. Gaussian case Impulses are isolated events over symbol period Wireless Networking and Communications Group

Wiener Filtering 43 Optimal in mean squared error sense in presence of Gaussian noise Model z(n) d(n) ^ ^ x(n) Design x(n) d(n) w(n) d(n) ^ w(n) d(n) e(n) Wireless Networking and Communications Group d(n): e(n): w(n): x(n): z(n): desired signal filtered signal error Wiener filter corrupted signal noise Minimize Mean-Squared Error E { |e(n)|2 } Return

Wiener Filter Design 44 Infinite Impulse Response (IIR) Finite Impulse Response (FIR) Wiener-Hopf equations for order p-1 Wireless Networking and Communications Group Return desired signal: d(n) power spectrum: (e j ) correlation of d and x: rdx(n) autocorrelation of x: rx(n) Wiener FIR Filter: w(n) corrupted signal: x(n) noise: z(n)

Results: Wiener Filtering 45 100 -tap FIR Filter Return Raised Cosine Pulse Shape n Transmitted waveform corrupted by Class A interference n Pulse shape 10 samples per symbol 10 symbols per pulse Channel A = 0. 35 = 0. 5 × 10 -3 SNR = -10 d. B Memoryless Received waveform filtered by Wiener filter n Wireless Networking and Communications Group

Filtering for Alpha Stable Noise 46 Myriad Filtering Sliding window algorithm outputs myriad of a sample window Myriad of order k for samples x 1, x 2, …, x. N [Gonzalez & Arce, 2001] As k decreases, less impulsive noise passes through the myriad filter As k→ 0, filter tends to mode filter (output value with highest frequency) Empirical Choice of k [Gonzalez & Arce, 2001] Developed for images corrupted by symmetric alpha stable impulsive noise Wireless Networking and Communications Group

Filtering for Alpha Stable Noise (Cont. . ) 47 Myriad Filter Implementation Given a window of samples, x 1, …, x. N, find β [xmin, xmax] Optimal Myriad algorithm 1. Differentiate objective function polynomial p(β) with respect to β 2. Find roots and retain real roots Evaluate p(β) at real roots and extreme points Output β that gives smallest value of p(β) 3. 4. Selection Myriad (reduced complexity) � � Use x 1, …, x. N as the possible values of β Pick value that minimizes objective function p(β) Wireless Networking and Communications Group

MAP Detection for Class A 48 Hard decision Bayesian formulation [Spaulding & Middleton, 1977] Equally probable source Wireless Networking and Communications Group Return

MAP Detection for Class A: Small Signal Approx. 49 Expand noise PDF p. Z(z) by Taylor series about Sj = 0 (j=1, 2) Approximate MAP detection rule We use 100 terms of the series expansion for d/dxi ln p. Z(xi) in simulations Logarithmic non-linearity + correlation receiver Near-optimal for small amplitude signals Wireless Networking and Communications Group Correlation Receiver Return

Incoherent Detection 50 Baye’s formulation [Spaulding & Middleton, 1997, pt. II] Small Signal Approximation Wireless Networking and Communications Group Correlation receiver Return

Filtering for Alpha Stable Noise (Cont. . ) 51 Hole Punching (Blanking) Filters Set sample to 0 when sample exceeds threshold [Ambike, 1994] Return Large values are impulses and true values can be recovered Replacing large values with zero will not bias (correlation) receiver for two-level constellation If additive noise were purely Gaussian, then the larger the threshold, the lower the detrimental effect on bit error rate Communication performance degrades as constellation size (i. e. , number of bits per symbol) increases beyond two Wireless Networking and Communications Group

MAP Detection for Alpha Stable: PDF Approx. 52 SαS random variable Z with parameters , , can be written Z = X Y½ [Kuruoglu, 1998] X is zero-mean Gaussian with variance 2 Y is positive stable random variable with parameters depending on PDF of Z can be written as a mixture model of N Gaussians [Kuruoglu, 1998] Mean can be added back in Obtain f. Y(. ) by taking inverse FFT of characteristic function & normalizing Number of mixtures (N) and values of sampling points (vi) are tunable parameters Wireless Networking and Communications Group Return

Results: Alpha Stable Detection 53 Return Wireless Networking and Communications Group

Complexity Analysis for Alpha Stable Detection 54 Return Method Complexity per symbol Analysis Hole Puncher + Correlation Receiver O(N+S) A decision needs to be made about each sample. Optimal Myriad + Correlation Receiver O(NW 3+S) Due to polynomial rooting which is equivalent to Eigen-value decomposition. Selection Myriad + Correlation Receiver O(NW 2+S) Evaluation of the myriad function and comparing it. MAP Approximation O(MNS) Evaluating approximate pdf (M is number of Gaussians in mixture) Wireless Networking and Communications Group

Performance Bounds (Single Antenna) 55 Channel Capacity Return System Model Case I Shannon Capacity in presence of additive white Gaussian noise Case II (Upper Bound) Capacity in the presence of Class A noise Assumes that there exists an input distribution which makes output distribution Gaussian (good approximation in high SNR regimes) Case III (Practical Case) Capacity in presence of Class A noise Assumes input has Gaussian distribution (e. g. bit interleaved coded modulation (BICM) or OFDM modulation [Haring, 2003]) Wireless Networking and Communications Group

Performance Bounds (Single Antenna) 56 Channel Capacity in presence of RFI Return System Model Capacity Parameters A = 0. 1, Γ = 10 -3 Wireless Networking and Communications Group

Performance Bounds (Single Antenna) 57 Probability of error for uncoded transmissions Return [Haring & Vinck, 2002] BPSK uncoded transmission One sample per symbol A = 0. 1, Γ = 10 -3 Wireless Networking and Communications Group

Performance Bounds (Single Antenna) 58 Chernoff factors for coded transmissions PEP: Pairwise error probability N: Size of the codeword Chernoff factor: Equally likely transmission for symbols Wireless Networking and Communications Group Return

Extensions to MIMO systems 59 RFI Modeling Middleton Class A Model for two-antenna systems [Mc. Donald & Blum, 1997] Closed form PDFs for M x N MIMO system not published Prior Work Much prior work assumes independent noise at antennas Performance analysis of standard MIMO receivers in impulsive noise [Li, Wang & Zhou, 2004] Space-time block coding over MIMO channels with impulsive noise [Gao & Tepedelenlioglu, 2007] Wireless Networking and Communications Group

Our Contributions 60 2 x 2 MIMO receiver design in the presence of RFI [Gulati et al. , Globecom 2008] RFI Modeling • Evaluated fit of measured RFI data to the bivariate Middleton Class A model [Mc. Donald & Blum, 1997] • Includes noise correlation between two antennas Parameter Estimation • Derived parameter estimation algorithm based on the method of moments (sixth order moments) Performance Analysis • Demonstrated communication performance degradation of conventional receivers in presence of RFI • Bounds on communication performance [Chopra et al. , submitted to ICASSP 2009] Receiver Design • Derived Maximum Likelihood (ML) receiver • Derived two sub-optimal ML receivers with reduced complexity Wireless Networking and Communications Group Backup

Performance Bounds (2 x 2 MIMO) 61 Channel Capacity [Chopra et al. , submitted to ICASSP 2009] Return System Model Case I Shannon Capacity in presence of additive white Gaussian noise Case II (Upper Bound) Capacity in presence of bivariate Middleton Class A noise. Assumes that there exists an input distribution which makes output distribution Gaussian for all SNRs. Case III (Practical Case) Capacity in presence of bivariate Middleton Class A noise Assumes input has Gaussian distribution Wireless Networking and Communications Group

Performance Bounds (2 x 2 MIMO) 62 Channel Capacity in presence of RFI for 2 x 2 MIMO Return [Chopra et al. , submitted to ICASSP 2009] System Model Capacity Parameters: A = 0. 1, G 1 = 0. 01, G 2 = 0. 1, k = 0. 4 Wireless Networking and Communications Group

Performance Bounds (2 x 2 MIMO) 63 Probability of symbol error for uncoded transmissions Return [Chopra et al. , submitted to ICASSP 2009] Pe: Probability of symbol error S: Transmitted code vector D(S): Decision regions for MAP detector Equally likely transmission for symbols Parameters: A = 0. 1, G 1 = 0. 01, G 2 = 0. 1, k = 0. 4 Wireless Networking and Communications Group

Performance Bounds (2 x 2 MIMO) 64 Chernoff factors for coded transmissions [Chopra et al. , submitted to ICASSP 2009] Parameters: G 1 = 0. 01, G 2 = 0. 1, k = 0. 4 Wireless Networking and Communications Group PEP: Pairwise error probability N: Size of the codeword Chernoff factor: Equally likely transmission for symbols Return

Results: RFI Mitigation in 2 x 2 MIMO 65 Receiver Quadratic Forms Exponential Comparisons Complexity Analysis for decoding M-QAM modulated signal Gaussian ML M 2 0 0 Optimal ML 2 M 2 0 Sub-optimal ML (Four-Piece) 2 M 2 0 3 M 2 Sub-optimal ML (Two-Piece) 2 M 2 0 M 2 Complexity Analysis Communication Performance (A = 0. 1, 1= 0. 01, 2= 0. 1, k = 0. 4) Wireless Networking and Communications Group

Extensions to Multicarrier Systems 66 Impulse noise with impulse event followed by “flat” region Coding may improve communication performance In multicarrier modulation, impulsive event in time domain spreads out over all subcarriers, reducing the effect of impulse Complex number (CN) codes [Lang, 1963] Unitary transformations Gaussian noise is unaffected (no change in 2 -norm Distance) Orthogonal frequency division multiplexing (OFDM) is a special case: Inverse Fourier Transform [Haring 2003] As number of subcarriers increase, impulsive noise case approaches the Gaussian noise case. Wireless Networking and Communications Group Return

Future Work 67 Modeling RFI to include Computational platform noise Co-channel interference Adjacent channel interference Multi-input multi-output (MIMO) single carrier systems RFI modeling and receiver design Multicarrier communication systems Coding schemes resilient to RFI Circuit design guidelines to reduce computational platform generated RFI Backup Wireless Networking and Communications Group